Introduction
2024-11-08
Answer: -136.7
Define a variable for each construct of the VAST display. These can be measured variables or unmeasured (mediating) variables.
You can either refer to an actual measurement procedure or simply define a variable. In both cases you should explicitly define the following properties:
In the Google doc, below your Construct Source Table, create a new table with the following columns:
Example:
| Construct in VAST display | Shortname | Type | Range/values | Anchors |
|---|---|---|---|---|
| Affective tone of instruction | aff_tone | Continuous | [-1; 1] | -1 = maximally negative 0 = neutral +1 = maximally positive |
| Anxiety | anxiety | Continuous | [0; 1] | 0 = no anxiety 1 = maximal anxiety |
| Kohlberg’s Stages of Moral Development | moral_stage | Ordinal | {1; 2; 3} | 1=Pre-conventional 2=Conventional 3=Post-Conventional |
| … | … | … | … |
Note: This resembles a codebook; but for theoretical variables, not for measured variables.
We want to model the following phenomenon (a specific version of the bystander effect):
Task: Sketch a first functional relationship that could model this phenomenon. Use the variables you defined in the previous step (including their labels and ranges).
Every causal path needs to be implemented as a mathematical function, where the dependent variable/output \(y\) is a function of the input variable(s) \(x_i\).
\(y = f(x_1, x_2, ..., x_i)\)
This can, for example, be a linear function, \(y = \beta_0 + \beta_1x_1\).
\(\color{red} y = \color{forestgreen} \beta_0 \color{black} + \color{forestgreen} \beta_1 \color{blue} x\)
→ \(\color{red} y\) = output variable, \(\color{forestgreen} \beta\)s = parameters, \(\color{blue} x\) = input variable.
Two types of parameters:
Note
Virtually all parameters (except natural constants) could be imagined as being free. It is a choice to fix some of them in order to simplify the model.
Fixing a parameter:
\(\color{forestgreen} \beta_0 \color{black} = 1 \rightarrow \color{red} y = \color{forestgreen} 1 \color{black} + \color{forestgreen} \beta_1 \color{blue} x\)
That means, the slope \(\color{forestgreen} \beta_1\) still can vary, but the intercept is fixed to 1.
Free parameters give flexibility to your function: If you are unsure about the exact relationship between two variables, you can estimate the best-fitting parameters from the data.
For example, sometimes a theory specifies the general functional form of a relationship (e.g., “With increasing \(x\), \(y\) is monotonously decreasing”), but does not tell how fast this decrease happens, where \(y\) starts when \(x\) is minimal, etc. These latter decisions are then made by the free parameters.
As a linear function is unbounded, it can easily happen that the computed \(y\) exceeds the possible range of values.
If \(y\) has defined boundaries (e.g., \([0; 1]\)), a logistic function can bound the values between a lower and an upper limit (in the basic logistic function, between 0 and 1):
\(y = \frac{1}{1 + e^{-x}}\)
With the 4PL* model from IRT, you can adjust the functional form to your needs, by:
*4PL = 4-parameter logistic model
(basic logistic function as dotted grey line)
The d, a, min, and max parameters can be used to “squeeze” the S-shaped curve into the range of your variables. For example, if your \(x\) variable is defined on the range \([0; 1]\), the following function parameters lead to a reasonable shift:
Of course, the logistic function and the beta distribution are just a start - you can use the full toolbox of mathematical functions to implement your model!
Note
These considerations about functional forms, however, are typically not substantiated by psychological theory or background knowledge - at least at the start of a modeling project. We choose them, because we are (a) acquainted to it, and/or (b) they are mathematically convenient and tractable.
Empirical evidence can inform both your choice of the functional form, and, in a model-fitting step, the values of the parameters.
my_functionarg1, arg2return(return_variable). If no explicit return() statement is given, the last evaluated expression is returned by default.Tips:
roxygen2 documentation standard)R function implements exactly one functional relationship of your model.#' Compute the updated expected anxiety
#'
#' The expected anxiety at any given moment is a weighted average of
#' the momentary anxiety and the previous expected anxiety.
#'
#' @param momentary_anxiety The momentary anxiety, on a scale from 0 to 1
#' @param previous_expected_anxiety The previous expected anxiety, on a scale from 0 to 1
#' @param alpha A factor that shifts the weight between the momentary anxiety (alpha=1)
#' and the previous expected anxiety (alpha=0).
#' @return The updated expected anxiety, as a scalar on a scale from 0 to 1
get_expected_anxiety <- function(momentary_anxiety, previous_expected_anxiety, alpha=0.5) {
momentary_anxiety*alpha + previous_expected_anxiety*(1-alpha)
}roxygen2 comments start with #' and are placed directly above the function definition.
@param parameter_name Description. Provide the range of possible values if applicable.@return DescriptionCheck out roxygen2 and document our exponential decay function with:
R with proper roxygen2 documentation.Connect all functions to one “super-function”, which takes all exogenous variables as input and computes the focal output variable(s).
Test the super-function:
We can tune our free parameters to fit the model as good as possible to empirical data. This is called model fitting.
We use the empirical data in
Which input variables do your simulated participants bring to the experiment? Real participants bring variability in these variables.
Assume a specific distribution of all input variables that participants bring to the experiment. Even better: find empirical evidence for it, for example from existing open data sets or summary statistics from publications.
Maybe you need to transform the scale of the empirical variable into the scale of your simulated equivalent (e.g., via z-standardization).
If you start simulating data for your virtual participants, you want to simulate their starting values yb drawing random values from a distribution. For example, the virtual participants might differ in their anxiety, which you previously defined on the range \([0; 1]\).
How can you generate random values that roughly look like a normal distribution, but are bounded to the defined range?
For simulations, it is good to know some basic distributions. Here are three interactive resources for choosing your distribution:
A handy distribution for the \([0; 1]\) range is the beta distribution. With its two parameters \(\alpha\) (also called \(a\) or shape1) and \(\beta\) (also called \(b\) or shape2), it can take many different forms:
How to choose \(\alpha\) and \(\beta\)? Asking ChatGPT/Wolfram Alpha for assistance
Assume that you intuitively started with a normal distribution with \(M=0.2\) and \(SD=0.1\) (rnorm(mean=0.2, sd=0.1)) to simulate your values. But this function can generate values < 0 and > 1.
How can you achieve a beta distribution that approximates the properties of this normal distribution? You can work out the math yourself (e.g., by looking at the formula for the mean and the variance of the beta distribution) - or you can ask ChatGPT. After all, we only use this function as a tool to get some plausible values.
“We have a normal distribution with mean=0.2 and SD=0.1. But the results should be bounded on a scale from 0 to 1. Create a beta-distribution that mimics the properties of the described normal distribution.”
“To mimic the properties of a specified normal distribution (with a mean and standard deviation) using a beta distribution within a bounded interval (in this case, 0 to 1), we need to find the parameters of the beta distribution (alpha \(\alpha\) and beta \(\beta\)) that match the mean and variance of the normal distribution as closely as possible.
[snip]
The parameters for the beta distribution that mimic the properties of the described normal distribution (with mean = 0.2 and standard deviation = 0.1, bounded between 0 and 1) are \(\alpha = 3\) and \(\beta = 12\).
This beta distribution should closely match the shape and spread of the specified normal distribution within the bounds of 0 to 1.”
You can generate random values in R with the rbeta function. Here’s a comparison of a normal distribution and a matched beta distribution that respects the boundaries \([0; 1]\):
Once all of these values are defined, simulate a sample of participants (with their starting values), and put half of them into the experimental group, and half of them into the control group. Use the resulting output values to computer a regular t-test.
Now the questions are:
Formal modeling in psychology - Empirisches Praktikum, Ludwig-Maximilians-Universität München